GOLDEN SECTION IN AMPHIPOLIS SPHINXES
L. Kaliambos (Natural Philosopher) January 31, 2016 In my paper “Discoveries in Amphipolis” I showed that Dinocrates in order to design the golden section of the two Caryatids in Amphipolis used the height α = 2.27 m of each Caryatid and the height β = 1.4 m of each pedestal. Indeed the Caryatids are on pedestals of height β = 1.4 m , making the total height (α + β) = 3.67 m of the statues. Dinocrates also designed the golden section in Amphipolis sphinxes which I discovered by using this reconstruction. Here we see that the greatest horizontal length of each sphinx (β = 1.267 m) is analogous to the base of a theoretical golden rectangle, while the total height ( α = 2.05 m) of each sphinx is analogous to the vertical side α of a theoretical golden rectangle. In mathematics 'GOLDEN SECTION '''is the division of a line segment into extreme and mean ratio. This is obtained by dividing a line into two parts such that the square of the one part is equal to the product of the whole segment and the other part. An approximate value for the ratio of the longer part (α) to the shorter part (β) is 1.62. Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities α and β with α > β >0, ( α +β ) / α = α/β = Φ = 1.61803398875... Where the Greek letter Φ represents the number of the golden ratio. Usually an approximate value of the ratio α/β is 1.62. That is, for the Caryatids we got (α + β)/α = α/β = (2.27 + 1.4) /2.27 = 2.27/1.4 =1.62 . Note that detailed measurements should be made by the excavation team could give the number Φ. For example writing b =1.402937 one could get (2.27+1.402937)/2.27 = 2.27/1.402937 = 1.618034 Using the same method of heights α = 8 m and β = 5m of the original lion located on the top of the mathematical tomb of hero Hephaestion Dinocrates planed also the golden ration of the lion. ( See my “Golden section in Ampipolis lion”). Such discoveries led also to the conclusion that the Amphipolis tomb is the funeral monument constructed for the divine hero HEPHAESTION and it is just the miniature of ancient Alexandria having the secrets of the Amun Oracle used in the ancient astronomy. In fact, after a careful analysis in a combinatory method similar to that of the British architect Ventris I discovered that the total height h = (α + β) of the lion was not (8+5 = 13m) but h = (α +β) = 1/12 stades × 157.5 = 13.125 m Under this condition Dinocrates in detail used α = 8.111696 and β = 5.013304. That is h = α +β = 8.111696 + 5.013304 = 13.125 m = 1/12 stades. Therefore (α +β)/α = α/β = Φ = (1 + 50.5)/2 = 1.618034 On the other hand for α = Φ and β =1 we may write (Φ + 1)/Φ = Φ/1 or Φ2 -Φ -1 = 0 and solving for Φ we get Φ = (1+50.5)/2 It means that Dinocrates for constructing the mathematical tomb of hero Hephaestion used algebra involving not only the mystic numbers of the foundation of Alexandria (331 BC) but also the mystic number Φ = (1+50.5)/2 of the oracle of Amun (323 BC). On the other hand for planning the golden section of the two sphinxes in Amphipolis Dinocrates used the well known method of golden rectangles used by Phidias in the construction of Parthenon ( See "The Parthenon and Phi the golden ratio") Throughout history, the ratio for length to width of golden rectangles has been considered the most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The Greeks were thought by some to have based the design of the Parthenon on this proportion, but this was subject to some conjecture. Nevertheless in my paper “Discovery of golden section in Parthenon” I showed that Phidias (500 BC – 432 BC), a Greek sculptor and mathematician, studied Φ and applied it to the design of sculptures for the Parthenon. Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos. In nature this golden section as a principle may be observed in the arrangements of leaves on a twig, petals on a flower and the arms of the starfish. The ancient Greeks considered a rectangle whose sides are in this ratio to be aesthetically the most pleasing of all rectangles and constructed their buildings on this principle. One of the interesting properties of the golden rectangle is that if you cut off a square section whose side is equal to the shortest side the piece that remains is also a golden rectangle. The golden rectangle has been known since antiquity as one heaving a pleasing shape, and is frequently found in art and architecture as a rectangular shape that seems ‘right’ to the eye. It is mentioned in Euclid’s Elements and was known to artists and philosophers. '''DINOCRATES BASED ON A GOLDEN RECTANGLE DESIGNED THE GOLDEN SECTION OF SPHINXES IN AMPHIPOLIS ' Ihis photo is from my interview I gave to the author of the Spiritual Thessaly, Mrs Dimitra Bardani, about the two sphinxes in Egypt which are similar to the two sphinxes in Amphipolis having the math of golden section. In a golden rectangle of height α and base of length β the golden section is given by (α +β)/α = α/β = Φ or for α = Φ and β =1 (unit length) we may write (Φ + 1)/Φ = Φ/1 or Φ + 1 = Φ2 or Φ2 -Φ -1 = 0 Then solving for Φ we get Φ = (1 +50.5)/2 = 1.618 In this case the golden rectangle of height α and base β behaves like a frame of a golden photo. That is, the golden photo of each sphinx is characterized by a golden rectangle of height α and base β. According to the “Amphipolis - The chronicle of the great excavation” on August 10, 2014 the excavations have revealed in the interior of the monument on the lintel of the gate of the tomb, two sphinxes - guards. They are headless with cut wings and traces of red paint on the feet, weighing 1.5 tons each. Note that later (Oct 21 , 2014) another amazing discovery was announced by the Amphipolis dig. It was the marble head of one of the sphinxes. The head was discovered whole with minimum fractures on the nose and it fits the eastern sphinx's torso, to which it was inserted. In the lower part of the throat, one can see the insertion incision. It has curly hair with traces of red, falling onto its left shoulder. The head was found in a 15 cm depth inside a marble threshold. It was a sculpture of exquisite art, leaving archaeologists in awe. Fractures of the sphinx's wings have also been located in the area. On the other hand according to the “ Amphipoli news: Αμφίπολη: Tο χρονικό” the total height of sphinxes is α = 2.05 m which is analogous to the vertical side α of a theoretical golden rectangle. Following the discovery of the marble head and portions of the wings of one of the damaged Sphinxes guarding the entrance of the Amphipolis tomb many have sought to envisage the statues as they were when first created. Now a blogger has succeeded in in creating one of the most convincing efforts to date. Panagiotis Krouklidis on his personal Italian blog has uploaded photoshopped images depicting the Sphinxes complete with wings, heads and their (assumed) red hair. Showing images of other ancient artifacts depicting Sphinxes, Mr Krouklidis makes a convincing case that his depictions are very close to the Sphinxes original incarnations. (See “Reconstructing the Sphinxes of Amphipolis”). Moreover in the “Reconstructing the Sphinxes of Amphipolis” we see that the greatest horizontal length of each sphinx (β = 1.267 m) is analogous to the base of a theoretical golden rectangle of the photo. Under this condition we write (α +β)/α = α/β = (2.05 + 1.267)/2.05 = 2.05/1.267 = Φ = (1+50.5)/2 = 1.618 . It is of interest to note that the British author Andrew wrote that a similar pair of female sphinxes was found by Mariette at the Serapeum at Saqqara near Memphis dated to the reign of the first Ptolemy by Lauer & Picard, mainly on the basis of an associated inscription: the Serapeum at Saqqara is also a strong candidate for the site of the first tomb of Alexander the Great.( See my "Alexander's worship in Amphipolis").Category:Fundamental physics concepts